Writing Linear Equations Linear equations can be represented in three ways : 1. Standard Form 𝑎𝑥+𝑏𝑦=𝑐 - where 𝑎 and 𝑏 are not both zero

Writing Linear Equations Linear equations can be represented in three ways : 1. Standard Form 𝑎𝑥+𝑏𝑦=𝑐 - where 𝑎 and 𝑏 are not both zero An example would be 4𝑥−3𝑦=12 𝑎=4 , 𝑏=−3 , 𝑐=12

Writing Linear Equations Linear equations can be represented in three ways : 1. Standard Form 𝑎𝑥+𝑏𝑦=𝑐 - where 𝑎 and 𝑏 are not both zero An example would be 4𝑥−3𝑦=12 𝑎=4 , 𝑏=−3 , 𝑐=12 2. Point – Slope Form 𝑦−𝑏=𝑚(𝑥−𝑎) - where 𝑚 is the slope, and ( 𝑎 , 𝑏 ) is a point on the line

Writing Linear Equations Linear equations can be represented in three ways : 1. Standard Form 𝑎𝑥+𝑏𝑦=𝑐 - where 𝑎 and 𝑏 are not both zero An example would be 4𝑥−3𝑦=12 𝑎=4 , 𝑏=−3 , 𝑐=12 2. Point – Slope Form 𝑦−𝑏=𝑚(𝑥−𝑎) - where 𝑚 is the slope, and ( 𝑎 , 𝑏 ) is a point on the line An example would be 𝑦−2=−2(𝑥+5) 𝑚=−2, point (2 , −5 )

Writing Linear Equations Linear equations can be represented in three ways : 1. Standard Form 𝑎𝑥+𝑏𝑦=𝑐 - where 𝑎 and 𝑏 are not both zero An example would be 4𝑥−3𝑦=12 𝑎=4 , 𝑏=−3 , 𝑐=12 2. Point – Slope Form 𝑦−𝑏=𝑚(𝑥−𝑎) - where 𝑚 is the slope, and ( 𝑎 , 𝑏 ) is a point on the line An example would be 𝑦−2=−2(𝑥+5) 𝑚=−2, point (2 , −5 ) 3. Slope – intercept Form 𝑦=𝑚𝑥+𝑏 - where 𝑚 is the slope, and 0 , 𝑏 is the y - intercept

Writing Linear Equations Linear equations can be represented in three ways : 1. Standard Form 𝑎𝑥+𝑏𝑦=𝑐 - where 𝑎 and 𝑏 are not both zero An example would be 4𝑥−3𝑦=12 𝑎=4 , 𝑏=−3 , 𝑐=12 2. Point – Slope Form 𝑦−𝑏=𝑚(𝑥−𝑎) - where 𝑚 is the slope, and ( 𝑎 , 𝑏 ) is a point on the line An example would be 𝑦−2=−2(𝑥+5) 𝑚=−2, point (2 , −5 ) 3. Slope – intercept Form 𝑦=𝑚𝑥+𝑏 - where 𝑚 is the slope, and 0 , 𝑏 is the y – intercept An example would be 𝑦=3𝑥+5 𝑚=3 , y – intercept is ( 0 , 5 )

Writing Linear Equations Standard Form and Point – Slope Form can both be written as Slope – Intercept Form with a little Algebra. Slope – Intercept Form is probably the most widely used format for linear equations / functions.

Writing Linear Equations Standard Form and Point – Slope Form can both be written as Slope – Intercept Form with a little Algebra. Slope – Intercept Form is probably the most widely used format for linear equations / functions. EXAMPLE : Find the slope and y – intercept for 2𝑥+3𝑦=−15

Writing Linear Equations Standard Form and Point – Slope Form can both be written as Slope – Intercept Form with a little Algebra. Slope – Intercept Form is probably the most widely used format for linear equations / functions. EXAMPLE : Find the slope and y – intercept for 2𝑥+3𝑦=−15 To write this equation in slope – intercept form, solve for 𝑦…

Writing Linear Equations Standard Form and Point – Slope Form can both be written as Slope – Intercept Form with a little Algebra. Slope – Intercept Form is probably the most widely used format for linear equations / functions. EXAMPLE : Find the slope and y – intercept for 2𝑥+3𝑦=−15 To write this equation in slope – intercept form, solve for 𝑦… 2𝑥+3𝑦=−15 −2𝑥 =−2𝑥 3𝑦=−2𝑥−15

Writing Linear Equations Standard Form and Point – Slope Form can both be written as Slope – Intercept Form with a little Algebra. Slope – Intercept Form is probably the most widely used format for linear equations / functions. EXAMPLE : Find the slope and y – intercept for 2𝑥+3𝑦=−15 To write this equation in slope – intercept form, solve for 𝑦… 2𝑥+3𝑦=−15 −2𝑥 =−2𝑥 3𝑦=−2𝑥−15 3𝑦 3 = −2𝑥 3 − 15 3

Writing Linear Equations Standard Form and Point – Slope Form can both be written as Slope – Intercept Form with a little Algebra. Slope – Intercept Form is probably the most widely used format for linear equations / functions. EXAMPLE : Find the slope and y – intercept for 2𝑥+3𝑦=−15 To write this equation in slope – intercept form, solve for 𝑦… 2𝑥+3𝑦=−15 −2𝑥 =−2𝑥 3𝑦=−2𝑥−15 3𝑦 3 = −2𝑥 3 − 15 3 𝑦= −2 3 𝑥−5

Writing Linear Equations Standard Form and Point – Slope Form can both be written as Slope – Intercept Form with a little Algebra. Slope – Intercept Form is probably the most widely used format for linear equations / functions. EXAMPLE : Find the slope and y – intercept for 2𝑥+3𝑦=−15 To write this equation in slope – intercept form, solve for 𝑦… 2𝑥+3𝑦=−15 −2𝑥 =−2𝑥 3𝑦=−2𝑥−15 3𝑦 3 = −2𝑥 3 − 15 3 𝑦= −2 3 𝑥−5 𝑚= −2 3 , y – intercept =( 0 , −5)

Writing Linear Equations EXAMPLE #2 : What is the slope and y – intercept for 𝑦−3=4(𝑥−2)

Writing Linear Equations EXAMPLE #2 : What is the slope and y – intercept for 𝑦−3=4(𝑥−2) Now, technically speaking, there is no Algebra needed. The slope is given in the format, 𝑚=4.

Writing Linear Equations EXAMPLE #2 : What is the slope and y – intercept for 𝑦−3=4(𝑥−2) Now, technically speaking, there is no Algebra needed. The slope is given in the format, 𝑚=4. And since the y – intercept always has an x – value of zero, we can substitute 𝑥=0 and solve for 𝑦.

Writing Linear Equations EXAMPLE #2 : What is the slope and y – intercept for 𝑦−3=4(𝑥−2) Now, technically speaking, there is no Algebra needed. The slope is given in the format, 𝑚=4. And since the y – intercept always has an x – value of zero, we can substitute 𝑥=0 and solve for 𝑦. 𝑦−3=4(0−2) 𝑦−3=4 −2 𝑦−3=−8 +3 =+3 𝑦=−5 y – intercept ( 0 , −5 )

Writing Linear Equations EXAMPLE #3 : What is the x – intercept for 3𝑥−5𝑦=−15

Writing Linear Equations EXAMPLE #3 : What is the x – intercept for 3𝑥−5𝑦=−15 x – intercepts occur where 𝑦=0. Substitute and solve for 𝑥.

Writing Linear Equations EXAMPLE #3 : What is the x – intercept for 3𝑥−5𝑦=−15 x – intercepts occur where 𝑦=0. Substitute and solve for 𝑥. 3𝑥−5 0 =−15 3𝑥=−15 3𝑥 3 = −15 3 𝑥=−5 x – intercept is ( −5 , 0 )

Writing Linear Equations How do we find the equation of a line given a graph ? It depends what you are given as your graph to work from.

Writing Linear Equations How do we find the equation of a line given a graph ? It depends what you are given as your graph to work from. This graph has a clear y – intercept and I can see the slope by counting using points.

Writing Linear Equations How do we find the equation of a line given a graph ? It depends what you are given as your graph to work from. This graph has a clear y – intercept and I can see the slope by counting using points. Y intercept is ( 0 , 4 ) 𝑚= 3 2 +2 +3 Slope – intercept form : 𝑦= 3 2 𝑥+4

Writing Linear Equations This graph does not have a clear y – intercept. But I can still identify some points, find the slope, and then use one of the points as 𝑎 , 𝑏 in point – slope form.

Writing Linear Equations This graph does not have a clear y – intercept. But I can still identify some points, find the slope, and then use one of the points as 𝑎 , 𝑏 in point – slope form. 𝑚= −4−(−1) 6−(−1) = −3 7 Using the point ( 6 , −4 ) and point – slope form : 𝑦− −4 = 3 7 𝑥−6 OR 𝑦+4= 3 7 (𝑥−6) ( −𝟏 , −𝟏 ) ( 𝟔 , −𝟒 )

Writing Linear Equations Find the equation of a line in slope - intercept form given two points on the line. Given points 2 , −1 , ( 3 , 3)

Writing Linear Equations Find the equation of a line in slope - intercept form given two points on the line. Given points 2 , −1 , 3 , 3 1. Find the slope 𝑚= 3−(−1) 3−2) = 4 1

Writing Linear Equations Find the equation of a line in slope - intercept form given two points on the line. Given points 2 , −1 , 3 , 3 Find the slope 𝑚= 3−(−1) 3−2) = 4 1 Use the point – slope form with 𝑚=4 and the point ( 3 , 3 ) 𝑦−3=4(𝑥−3)

Writing Linear Equations Find the equation of a line in slope - intercept form given two points on the line. Given points 2 , −1 , 3 , 3 Find the slope 𝑚= 3−(−1) 3−2) = 4 1 Use the point – slope form with 𝑚=4 and the point ( 3 , 3 ) 𝑦−3=4(𝑥−3) Solve for 𝑦 𝑦−3=4𝑥−12 +3= +3 𝑦=4𝑥−9 𝑚=4 , y – intercept is ( 0 , −9 )

Writing Linear Equations EXAMPLE : The temperature of a liquid cools 4° F every minute. After 5 minutes, the temperature of the liquid is 175° F. Which equation best models the temperature, 𝑦, of this liquid after 𝑥 minutes ? A) 𝑦=−4𝑥+155 C) 𝑦=−5𝑥+155 B) 𝑦=−4𝑥+195 D) 𝑦=−5𝑥+195

Writing Linear Equations EXAMPLE : The temperature of a liquid cools 4° F every minute. After 5 minutes, the temperature of the liquid is 175° F. Which equation best models the temperature, 𝑦, of this liquid after 𝑥 minutes ? A) 𝑦=−4𝑥+155 C) 𝑦=−5𝑥+155 B) 𝑦=−4𝑥+195 D) 𝑦=−5𝑥+195 SOLUTION : The slope 𝑚=−4 because of the liquid cooling 4° F every minute. It is a rate of change…

Writing Linear Equations EXAMPLE : The temperature of a liquid cools 4° F every minute. After 5 minutes, the temperature of the liquid is 175° F. Which equation best models the temperature, 𝑦, of this liquid after 𝑥 minutes ? A) 𝑦=−4𝑥+155 C) 𝑦=−5𝑥+155 B) 𝑦=−4𝑥+195 D) 𝑦=−5𝑥+195 SOLUTION : The slope 𝑚=−4 because of the liquid cooling 4° F every minute. It is a rate of change… We can use the point ( 5 , 175 ) as a point on the line. 𝑦−𝑏=𝑚 𝑥−𝑎 𝑦−175=−4(𝑥−5) 𝑦−175=−4𝑥+20 +175= +175 𝑦=−4𝑥+195 So ANSWER is B